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The other thread brought to my mind another fascinating accent question I recently ran into. It concerns the accentuation of περι in Epic when it's used as an adverb with the meaning "exceedingly".

LSJ, περί E.II.1. says it "commonly suffers anastrophe", e.g. Τυδεΐδη, πέρι μέν σε τίον Δαναοί Il.8.161. This and the examples in Homer that LSJ gives in this section are, however, accented περί in the edition that TLG provides, as well as in West's Iliad and van Thiel's Odyssey; i.e., everybody seems to ignore this rule.

However, in the next section (E.II.2.) LSJ states that περὶ κῆρι does not suffer anastrophe.

The newest paper edition of LSJ isn't different in this respect from the online version.

I wonder what basis LSJ has for these principles, or what basis the editors have to deviate from it.

There are a number of small points of accentuation on which the sources--the Byzantine manuscripts on which our texts are based--follow divergent practices. This is probably one of them. You might check West's Iliad--his apparatus is very careful and precise in recording the evidence of the mss. on points like this.

You're right, I only looked in West's preface where he discusses some orthography issues, and not the most obvious place, the apparatus. I checked a couple of places, and περί is the majority reader in them, in medieval manuscripts as well as in someone called Herodianus, who apparently was a scholiast of some sort.

Aelius Herodian was a grammarian who lived in the 2d century CE during the era of Marcus Aurelius. He is the ultimate source of much of the information we have about ancient Greek accentuation (and perhaps some misinformation, too).

See Probert, A New Short Guide to the Accentuation of Ancient Greek, secs. 21-2, pp. 13-14.

Herodian wrote several treatises on accents, which are lost but are known through an epitome written around 400 attributed to Arcadius or Theodosius, some later Byzantine grammarians, and the A scholia to the Iliad (those found in the Venetus A ms.), which are believed to incorporate material from a commentary that drew on his work among that of others (the so-called Viermännerkommentar, or VMK, "four-man commentary"; Herodian was one of the four men).

So our information about Greek accents has come down through the ms. tradition, as well as in grammatical works and scholia derived from Herodian. I'm sure some inaccuracies and misconceptions must have crept into this material, especially about very minor points, especially since the first-hand material we have dates from the era after the tone accentuation of ancient Greek had given way to a stress accent. And there are some disputed topics, too.

Addendum: you might find this Wikipedia article on the Venetus A ms. interesting:

Thanks. You are, again, doing all the work for me. I was familiar with the concept of Viermännerkommentar, but had no recollection that Herodian was one of them. I've read (parts of) stuff like West's Text and transmission of the Iliad, but it's difficult to remember so many names, as long as they are only names to me.

West, in the same passage in the preface to his Iliad (p. xix) as that which you cited in the other thread, says that he agrees with Herodian and Aristarchus, who maintained, against Tyrannio and Ptolemy, that περί doesn't undergo anastrophe when it means περισσῶς (i.e., "exceedingly"). West writes that the difference [between περί and πέρι, I guess] isn't in the meaning but in the syntax.